## Section: New Results

Keywords : Time substepping, fluid-solid coupling, high-order finite elements.

### Improved numerical approaches for seismic wave equation at a fluid-solid interface in the oil industry

Participants : Ronan Madec, Dimitri Komatitsch, Julien Diaz, Pierre Thore.

In the oil industry, many important oil reservoirs are located offshore and it is therefore of interest to be able to simulate seismic wave propagation in deep offshore geological media and it is crucial to reduce the cost of the calculations in the thick but homogeneous and therefore simple water layer. Explicit numerical methods used to model wave propagation must satisfy a stability condition called the Courant-Friedrichs-Lewy condition (CFL) that depends on the maximum velocity of the medium and on the ratio between the time step and the size of a grid cell. In the case of a deep offshore model, the velocity of P waves in the thick homogeneous water layer located upon the ocean bottom is generally slower than in the solid part of the model located underneath the ocean flow and therefore the local CFL is different within each medium. But to ensure numerical stability, one has to take a small global time step which is imposed by the solid part of the medium, and therefore waste some calculation time in the simpler fluid part of the model. An idea that can be used to overcome this is to implement substepping in time to reduce the computation time by taking a local time step and still honor the CFL condition by increasing the time step in the layer of slower velocity (the water layer). We therefore implemented in our spectral-element code SPECFEM3D the method developed in [55] that ensures the conservation of energy along the interface and we adapted it to a high order finite-element method. Taking advantage of the computation time that we save we can therefore start thinking about computing large and more realistic 3D models of interest for the oil industry in the next few months.

Another issue that we have studied in [18] is how to implement plane wave sources with any incidence angle for our spectral-element time-domain method. We also implement absorbing boundary conditions following the ideas of [49] , which is crucial to ensure that no significant spurious waves propagate back into the main domain. We validate the method and check its accuracy for the numerical modeling of seismic wave propagation, in particular in order to compute the response of a free surface with topography under the incidence of a plane wave with different angles by comparing the results obtained to the Method of Fundamental Solutions (MFS), which is a new method to solve the wave equation in the frequency domain.